1. Field of the Invention
The present invention relates generally to apparatus and methods of optical image processing.
2. Description of the Related Art
Various simple optical systems yield the Fourier transform (FT) of a two-dimensional complex object function. (See, e.g. J. W. Goodman, “Introduction to Fourier Optics,” McGraw-Hill, New York, 2002). Physical examples of such two-dimensional complex object functions include, but are not limited to, transparent objects such as photographic transparencies, spatial light modulators, and biological samples that modifies both the amplitude and the phase of transmitted (or reflected) optical waves. One additional example of such an optical system is simply the free-space propagation, far-field diffraction pattern (the Fraunhofer pattern). The Fraunhofer diffraction pattern yields the FT of the complex transmission function of an aperture that is illuminated with plane waves.
Another simple optical system that yields the FT of a two-dimensional complex object function is a thin converging lens. At the focal plane of the lens, the formed image is simply the FT of the object function placed anywhere before the image plane, preferably at the front focal plane. However, for both of the above-mentioned systems, only the FT magnitudes are detected, so direct phase measurement is a difficult task.
In other systems, femtosecond pulses have been extensively used in physics and chemistry to resolve fast transient response of various material properties. In many of these fields, the transient changes induced in the material properties due to the presence of a pump beam are of interest. To be able to record these fast transient effects, femtosecond spectral interferometry (SI) has been widely used. (See, e.g., F. Reynaud et al., “Measurement of phase shifts introduced by nonlinear optical phenomena on subpicosecond pulses,” Opt. Lett., Vol. 14, page 275 (1989); E. Tokunaga et al., “Frequency-domain interferometer for femtosecond time-resolved phase spectroscopy,” Opt. Lett., Vol. 17, page 1131 (1992); E. Tokunaga et al., “Induced phase modulation of chirped continuum pulses studied with a femtosecond frequency-domain interferometer,” Opt. Lett., Vol. 18, page 370 (1993); J. P. Geindre et al., “Frequency-domain interferometer for measuring the phase and amplitude of a femtosecond pulse probing a laser-produced plasma,” Opt. Lett., Vol. 19, page 1997 (1994); D. W. Siders et al., “Plasma-based accelerator diagnostics based upon longitudinal interferometry with ultrashort optical pulses,” IEEE Trans. Plasma Science, Vol. 24, page 301 (1996); C. W. Siders et al., “Laser wakefield excitation and measurement by femtosecond longitudinal interferometry,” Phys. Rev. Lett., Vol. 76, page 3570 (1996); R. Zgadzaj et al., “Femtosecond pump-probe study of preformed plasma channels,” J. Opt. Soc. Am. B, Vol. 21, page 1559 (2004); L. Lepetit et al., “Linear techniques of phase measurement by femtosecond spectral interferometry for applications is spectroscopy,” J. Opt. Soc. Am. B, Vol. 12, page 2467 (1995); S. M. Gallagher et al., “Heterodyne detection of the complete electric field of femtosecond four-wave mixing signals,” J. Opt. Soc. Am. B, Vol. 15, page 2338 (1998); J. Tignon et al., “Spectral interferometry of semiconductor nanostructures,” IEEE J. Quantum Electron., Vol. 35, page 510 (1999); X. Chen et al., “Temporally and spectrally resolved amplitude and phase of coherent four-wave-mixing emission from GaAs quantum wells,” Phys. Rev. B, Vol. 56, page 9738 (1997); D. Birkedal et al., “Femtosecond spectral interferometry of resonant secondary emission from quantum wells: Resonance Rayleigh scattering in the nonergodic regime,” Phys. Rev. Lett., Vol. 81, page 2372 (1998); C. Dorrer et al., “Spectral resolution and sampling issued in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B, Vol. 17, page 1795 (2000); C. Dorrer, “Influence of the calibration of the detector on spectral interferometry,” J. Opt. Soc. Am. B, Vol. 16, page 1160 (1999).)